
where \(c_P=\operatorname{tr}(PA)/2^n\)  are the coefficients of \(A\)  in the Pauli basis. Using simple operators \(\sigma _+ = |0\rangle \!\langle 1|\) , \(\sigma _- = |1\rangle \!\langle 0|\) , the number of terms in the decomposition can be reduced to \(2 \log N+1\)  [1]}. A more efficient approach, however, is to express \(A\)  as a linear combination of unitary transformations of simple Hamiltonians [2]}. Accordingly, the decomposition of \(A\)  in one dimension can be written as [3]}
\(A_{x, \beta }=I^{\otimes n-1} \otimes (I-X)+S^{\dagger }\left[I^{\otimes n-1} \otimes (I-X)+I_{0}^{\otimes n-1} \otimes \left(X-a_{\beta } I\right)\right] S,\) 
